On Topological Lower Bounds for Algebraic Computation Trees

Abstract

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set $$\Sigma \subset {\mathbb R}^n$$ is bounded from below by $$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\Sigma ))}{m+1} -c_2n, \end{aligned}$$ where $$\mathrm{b}_m(\Sigma )$$ is the mth Betti number of $$\Sigma $$ with respect to “ordinary” (singular) homology and $$c_1, c_2$$ are some (absolute) positive constants. This result complements the well-known lower bound by Yao (J Comput Syst Sci 55:36–43, 1997) for locally closed semialgebraic sets in terms of the total Borel–Moore Betti number. We also prove that if $$\rho :\> {\mathbb R}^n \rightarrow {\mathbb R}^{n-r}$$ is the projection map, then the height of any tree deciding membership in $$\Sigma $$ is bounded from below by $$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\rho (\Sigma )))}{(m+1)^2} -\frac{c_2n}{m+1} \end{aligned}$$ for some positive constants $$c_1, c_2$$ . We illustrate these general results by examples of lower complexity bounds for some specific computational problems.